Interesting Problems

From Umesh: I take a match stick and break into three parts. Sometimes
it is possible to arrange the parts into a triangle, but sometimes it is not
e.g. if one length is very long and the others are very short. If I break the
match at two randomly chosen positions, what's the
probability that the three parts can be arranged into a triangle? Is this
answer changed if I break the match at a random position, and then break the
longest of the two resulting parts at a random position?

From Tom Quilter: Can an irrational number raised to the power
of another irrational number ever be rational? Give an example -
you're not allowed to use complex numbers.

From Jon Middleton: You have a 1 metre long ruler and a collection of N ants. The ants, when placed on the ruler, will travel in a straight line at 1m per hour. If they meet another ant they change direction and go the other way. If they get to the end of the ruler they fall off. How long is it before all of the ants fall off the ruler? How does this depend on the intial locations and directions of the ants?

What are the following famous numbers, a = √(1 +
√(1+√(...))) or b = 1/(1+1/(1+1/...))?

A class contains N female students and N male students.

I put the class into pairs by placing all of the students' names into a hat and
drawing out their names one by one. The first two names pulled out of the hat are put into a pair, the next two form a pair, and so on until the hat is empty.

What's the probability that the paired up class contains
1) only same sex pairs?
2) only mixed sex pairs?

From Ed Barton: I hold a slinky by one end so it is suspended
(not touching the ground), stretched under gravity, and then let go -
describe the motion of the two ends.

A very clever person runs a rope all the way around the
equator. How much longer must he make the rope if he wants it to make
it 1 metre off the ground everywhere? Is this surprising?

A frog sits at the bottom of a flight of stairs. The frog can hop up either one stair or two stairs at a time. How many ways can the frog hop up N stairs? What famous series is this?

From Iain Murray: A mile long stretch of railway track, fixed at
both ends, expands by 1 foot in the sunshine. It bows in such away
that it runs along an arc of a circle. How high is the mid point off
the ground?

From Jeff Beck: A duck is at the centre of a circular pond, but wants to get to
the edge. A fox situated at the edge of the pond and can run round and
gobble the duck. How fast does the duck need to be, compared to the
fox, to reach a point on the edge of the pond before the fox?

If p is a prime number greater than three, explain why p2-1 is always divisible by 24 with no remainder.

I lay out 1000 coins in a row, heads facing upwards. I then flip every second coin so that they are tails up. I then flip every third coin. So now, for example, coin three is tails up having been flipped once whilst coin 6 is back to being heads up having been flipped twice. I repeat this process flipping every fourth, fith, etc. coin, up to 1000. The question is, at the end of this process, which coins remain heads up?

The Stamp Problem: You have two types of stamps worth A pence and B pence and you can put as
many as you like onto an envelope. The task is to find the expression for the
largest total value that it is impossible to make N(A,B). e.g. if A=7, B=5, the
largest unmakeable value is N=23.

The two envelopes paradox: You can pick one of two envelopes. Both of the
envelopes contain money, but one contains twice the amount of the other. You choose one, open it, and see it contains $Y. Should you change your choice of envelope?